Energy Sharing in the Random Vibration of Non-linearly Coupled Modes

If small non-linear terms couple the (linear) modes of a multidegree-of-freedomsystem, energy can flow from one mode to another and vice versa.In this paper a perturbation method is used to calculate theaverage rate of energy transfer between nonlinearly coupledmodes. Each mode is assumed to be subjected to an independentsource of stationary, Gaussian, random excitation, and the couplingterms, which are conservative, are of the form x_rx_s, and x_r, where x_r and x_s are the normal co-ordinates of the linearizedsystem. The principal result is that, if the excitation is whitenoise, no flow of energy occurs when each mode has the sameenergy. Thus, as predicted by the results of statistical mechanics,the equilibrium state for white noise excitation is that ofequipartition of energy between the modes. When the modal energiesare not equal, energy flows from modes of higher energy to modesof lower energy, and the rate of energy transfer is greatestfor those modes whose natural frequencies satisfy internal resonancecombinations. An application of the results of the paper is in calculatingnoise transmission between connected structures, where vibrationalmodes may be coupled by small nonlinear terms. If the naturalfrequencies of the coupled modes are suitably related, it isshown that very small coupling may cause considerable excitationof otherwise dormant parts of a structure. The paper may alsobe of interest in the study of molecular reactions and biologicalrhythms, and of non-linear wave interactions in such areas asplasma dynamics, turbulence, and the theory of water waves.